Purpose
This unit examines the properties of polygons and how these are related. It also gives the names in both Māori and English.
Specific Learning Outcomes
 Investigate properties of symmetry in shapes.
 Investigate spatial features of shapes.
 Use both English and Te Reo Māori to describe different polygonal shapes.
Description of Mathematics
This unit allows students to develop an understanding of the geometrical features of polygons and how classes of polygons are defined. It also aims to develop aspects of symmetry (reflective and rotational) through a problem solving approach.
A polygon is a planar (flat) shape that is bounded by straight sides. The relationships between sides and angles are used to create classes (groups) of shapes. In this unit students form quadrilaterals and triangles. Each class of shapes contains subclasses. For example, a quadrilateral might have pairs of opposite parallel sides. If a quadrilateral has only one pair of parallel sides, it is called a trapezium. If it has two pairs parallel sides, it is called parallelogram. Some parallelograms have internal angles that are right angles. That class of shapes is called rectangles. Definition, and reasoning with those definitions, is an important feature of geometric thinking.
In this unit, mathematical language is also explored particularly in terms of te reo Māori. It is envisaged that such an exploration will give rise to descriptions that can incorporate both languages, to support students to make sense of defining properties of 2dimensional and 3dimensional shapes.
The unit begins with string geometry to set the scene for investigating shapes and their properties using folding and possible turning techniques. Activity progresses to an examination of regular polygons where Māori terms are introduced. The concept of whānau or family is introduced to reinforce the fact that polygons are linked in a range of ways.
For this unit you will need to know, and the students will need to find out, the following:
porowhita = circle
whānau = family
taparau = polygons
tapatoru = triangle
tapawhā = square
tapawhā whakarara = parallelogram
whitianga = diameter
pūtoro = radius
paenga = circumference
tapatoru rite = equilateral triangle
puku = tummy
e toru, nga tapatoru rite = made from 3 smaller equilateral triangles
tapawhā whakarara rite = a rhombus
taparara = trapezium
tapawhā rite = square
tapawhā hāngai = an oblong
koeko tapatoru = a triangular pyramid
ahu3 = 3 dimensions
tapaono rite = regular hexagon
kai = food
e ono, tapatoru rite = 6 equilateral triangles
Opportunities for Adaptation and Differentiation
The learning opportunities in this unit can be differentiated by providing or removing support to students, or by varying the task requirements. Ways to support students include:
 ensuring that students have access to the physical manipulatives suggested, so they can experiment with forming and folding shapes, before being required to visualise transformations
 explicit modelling of forming shapes folding, reflections and rotations, where students need support. Expect students to copy your actions before attempting problems independently
 helping students to represent shape forming, reflections and rotations diagrammatically to ease memory load and support thinking, e.g. drawing lines of symmetry on a paper copy of a shape
 providing a list of mathematical terms, and definitions for students to refer to
 using the art of story telling to provide a motivational setting for mathematical inquiry.
Tasks can be varied in many ways including:
 beginning with simple shape problems with one or two conditions, before engaging students in tasks with multiple conditions
 using collaborative grouping, particularly with physical activity, so students can support others
 reducing the demands for a product, e.g. diagrams with less reliance on writing sentences
 altering the extent to which students need to visualise actions, or to reason abstractly with the properties of classes.
The contexts for this unit can be adapted to suit the interests and cultural backgrounds of your students. Capitalise on the interests of your students. Shape, especially those with symmetry, is a commonality across most human cultures. Look for shapes and symbols that embody the target two dimensional shapes. For example, tukutuku panels use the symmetries of shapes in design. Three dimensional shapes are common in the environment and in human construction. Whare are usually the shape of a prism and the properties of prisms give them structural strength. Pyramids are well known to students through studies of Egypt and Central American civilisations, and are common in playgrounds and modern architecture.
Required Resource Materials
 String or elastic to make 23 metre long loops
 Drawing compasses
 Paper and scissors
 Polyhedral models (optional)
Activity
Getting Started
In this session we use loops of string or elastic to form shapes according to given requirements. We then convince others that our shape meets the requirements.
 Put students into groups of 4 and give each group a length of string or elastic  approximately 2 to 3 meters long. Have them tie the ends of the string in order to make a loop.
 Pose the following task:
Use the members of your group and the loop of string to make a shape that has four equal sides.
 Allow the students to experiment, but provide questions that focus on the symmetry of the shape and the ‘squareness’ of the corners. For example:
How did you find each of the equal length sides?
Using the string, how might you show any lines of reflective (line) symmetry?
How might you show whether the corners are right angles (without using a protractor)?
Convince another team member that your shape is correct.
 Let students share their shapes before gathering the class.
What is the mathematical name for your shape?
Most groups will create squares, but some groups may create rhombi (rhombuses).
What properties must a shape have to be a square? (Four equal sides, four right angles, two pairs of parallel sides)
Are there other shapes that have four equal sides?
What are those shapes called? (Rhombi)
What are the properties of a rhombus? (“Four equal sides” is the only defining property. Note: Opposite angles are equal so there are two pairs of equal angles.)
Is a square a kind of rhombus? (Yes. It has four equal sides).
 You might draw a large Venn Diagram on the floor or whiteboard to illustrate the set and subset relationship of rhombi and squares. Ask students to draw squares and rhombi to add to the diagram. Make sure that they realise that the squares, inside the smaller circle, are also counted as rhombi, since they are also inside the larger circle.
Turn various shapes so they appear differently, and ask:
Is this still a square/rhombus? Why? (Orientation does not effect the properties of the shape.)
 Now challenge the groups to use the loop of string to complete this task:
Use the members of your group and the loop of string to make a shape that has four sides, two of which are parallel.
You may need to define parallel first before groups start work. Look up a definition online. Railway tracks are a good metaphor.
 After a suitable time, groups can share their shapes and justify that the shape they created meets the criteria. Most groups are likely to make parallelograms or rectangles, but some may form trapezia. Each class of shape is correct.
 Define each class of shapes, first by asking students about the properties they see, then providing them with a definition.
 Parallelogram: Quadrilateral with opposite sides that are parallel
 Rectangle: Quadrilateral with four right angles
 Trapezium: Quadrilateral with only one pair of opposite sides that are parallel.
Note: In the US, a trapezium refers to a quadrilateral with no parallel sides, and some definitions of trapezium refer to at least one pair of parallel sides, therefore including parallelograms.

Is a rectangle a special parallelogram? (Yes, all rectangles have two pairs of parallel sides, so they are all types of parallelograms).
Is a trapezium also a type of parallelogram? (A trapezium has only one pair of parallel sides, so it is not a parallelogram).
 Use your loop of string to make the following triangles:
 a triangle that has no sides that are the same (scalene)
 a triangle that has only 2 sides the same (isosceles)
 an equilateral triangle.
 After making each type of triangle, the group must prove that the necessary conditions are met. Though the conditions involve side length it is important to focus attention on angles. Use 90⁰ (a right angle) as a benchmark. Most triangles that students create have angles less than 90⁰.

Can you form triangle with an angle greater than ninety degrees? Try. (Yes, e.g. 120⁰, 35⁰, 25⁰)
What is true about the interior angles of an equilateral triangle? (Equal angles)
Can a triangle have one right angle? (Yes) Make a triangle like that.
Can a triangle have two right angles? (No)
 Ask the students to create a set of criteria for their own shape. The shape can have up to eight sides and there can be up to four criteria. Each group needs to provide a model answer to their teacher, with awareness that more than one class of shape might be possible.
 Gather as a class. Ask groups to provide their criteria while the other groups construct the shape with string or elastic.
Session 2
Over the next few sessions, students create Te Whānau Taparau (The Polygon Family) by transforming a circle.
The Circle
 This is a story about a shape from the porohita (circle) whānau (family) who wanted to be part of the taparau (polygon) whānau. He was a special porohita because he was able to transform himself into other shapes by folding and unfolding. However, he needed the help of a mathematician who knew some things about geometry ... that person is you!
 Either give the students a set of porohita, or give the students a piece of A4 paper and ask them to make the largest possible circle using drawing compasses. Discuss how the students worked out the biggest circle. If compasses are not available simply trace around a circular plate or lid.
 Discuss the attributes of a circle.
What makes a circle different from polygonal shapes, for example the triangle (tapatoru), square (tapawhā) or the parallelogram (whakarara)?

Students should note that polygons have straight sides. In fact, a polygon is a 2dimensional shape that is bounded by three or more straight sides.
 Is a circle a type of polygon? (No. A circle is bounded by a curved ‘side’.)
What are some names that are used to describe parts of a circle?
 Focus the students on:
 diameter (rangiwhitu), radius (putoro) and circumference (pae)
 the relationship between te rangiwhitu and te putoro. The radius equals half the diameter.

It is important for students to consider the idea of a circle as a locus, an infinite set of points that meet a condition or set of conditions. Take your students outside with skipping ropes and chalk. Challenge them in teams of three or four.
Use your rope and chalk to create a perfect circle.
 Students are likely to try many different techniques, but the most successful method is to emulate a tethered pony. One student fixes an end of the rope to spot (the centre), while another walks around them with the rope kept taut. That student traces their path with chalk.

The defining property of a circle is that it is a set of points the same distance from the centre. The length of the rope is the radius of the circle.
The relationship between pae and rangiwhitu is an extension or extra investigation, usually reserved for later levels. Pi (3.14…) is the ratio of circumference to diameter, that is the circumference equals the diameter multiplied by pi.
 Return to the class and challenge your students to:
Find the centre of your paper circle. What methods can you find?

Folding the circle in half in two different orientations creates an intersection of diameters. The point of intersection is the centre.
The circle becomes an equilateral triangle
 We now continue the activity to transform the circle:
Porowhita looked in the mirror one day and decided he was getting bored of being a circle. Porowhita decided that he wanted to look just like tapatoru rite. He needs your help.
Using the circles (provided or made), fold Porowhita to make the biggest possible tapatoru rite (equilateral triangle). You may need to experiment with different ways of folding to get the largest one possible.

Develop a method that you will use in order to check that the shape you have made satisfies the requirements needed for a tapatoru rite, and that it is the largest equilateral triangle possible.
 If this task is too challenging for the students, you may need to give a hint (refer to the diagram) or show one that has already been made, to convince students that it can be done.
 Focus the followup discussion on the rotational and reflective symmetry of the triangle.
How many lines of reflective symmetry does the triangle have?
Where are the lines located?
Does the triangle have rotational (turn) symmetry?
Why must the internal angles be equal for the shape to have rotational symmetry?

What other types of triangle can be folded from a circle?
What happens if you do not fold into the centre of the circle?
Is a rightangled triangle possible? (One side to the triangle must be the diameter)
 It would be worthwhile getting the students to develop methods to check for other tapatoru rite, and in so doing, develop the relationship between length of the sides and size of the angles.
Session 3
Triangle to trapezium
 So, Porowhita not only changes his shape, but also changes his name to Tapatoru rite (equilateral triangle). At the end of the week Tapatoru rite feels cramped  he can't roll around like he used to when he was a porowhita. His aching sides and corners needed to be massaged in order to get rid of some of his pain. While lying on his puku reading the Geometers’ Weekly magazine, he spots his favourite sports hero, Jonah Trapz who is ‘built like a taparara’ (trapezium). Immediately, Tapatoru rite wishes he could transform himself into a taparara. But he needs your help.
 As a possible way of introducing students to a trapezium, have them explore attributes of various trapezoidal shapes. Have them compare these with nontrapezoidal shapes including quadrilaterals that are parallelograms, rectangles, squares and others that are ‘nearly’ trapezoidal in shape. One approach is illustrated as follows:
Here is Hannah's work. She has sorted some quadrilateral shapes into two different categories: trapezia and not trapezia.
Trapezia

Not Trapezia



Hannah states that her sorting procedure is based on the relationship between one pair of sides. If just one pair of sides is parallel, then it’s a trapezium. The other pair of sides cannot be parallel.
Do the parallel sides have to be different lengths? Why? (If the side lengths are equal then the other pair of sides will be parallel. The quadrilateral will be a parallelogram not a trapezium)
 Ask the students to explore Hannah’s table of shapes and explain her sorting procedure.
Ask questions that encourage the students to focus on quadrilaterals that have one pair of parallel sides. Some students might observe that a trapezium often looks like a triangle with the top cut off leaving an edge that is parallel to the side opposite.
 Tapatoru rite wants to look like his sports hero Jonah Trapz.
Explore how you might fold Tapatoru rite to make a taparara (trapezium).
The trapezium must contain three equilateral triangles within it, that are the same size.

Let students work on the challenge in pairs.
The following diagram illustrates the change to a trapezium.
Using the taparara (trapezium) that you have made:
 How would you check that a pair of sides is parallel?
 How would you use your understanding about the equilateral triangle to help you make a convincing argument for a pair of parallel sides?
 How would you check that each of the smaller tapatoru rite are of equal size and shape, that is they are congruent?
Trapezium to rhombus
 To go with his new shape, Tapatoru rite changes his name to Taparara (trapezium). One day, he attempts to enter the Rhombus Exhibition at the Geometrical Museum  rhombi are one of Taparara’s favourite shapes. He is stopped at the door and told that only whakarara shapes (parallelograms) may go in. Taparara really wants to get in, so he decides to change shape. He doesn’t want to be just any whakarara, he wants to be a whakarara rite (rhombus). He needs your help.
 Ask the students to fold Taparara to make a whakarara rite. The following diagram illustrates the change from a trapezium to a rhombus:
 Using the whakarara rite made ask:
 How would you check that the sides are parallel?
 How would you check that all the sides are of equal length?
 What might the relationship be between the angle measure of the smaller corner and the angle measure of the larger corner? (Since the triangles are equilateral, the large interior angles of the rhombus are 60⁰ + 60⁰ = 120⁰)
 Of course, Taparara changes his identity and name and becomes known as Whakarara rite. Some of his friends call him Rhombus.
Write two sentences that describe Whakarara rite’s appearance.
What fraction of the original tapatoru rite (equilateral triangle) is Whakarara rite? (One half)
Session 4
Rhombus to a triangular pyramid
 Once in the Geometrical Museum, Whakarara rite meets up with some of his friends, Tapawhā rite (square) and her sister Tapawhā hāngai (oblong).
 At this point the students should do some personal research and write a report, using diagrams if needed, that illustrate the difference between the rhombus (whakarara rite), the square (tapawhā rite), and the oblong (tapawhā hāngai).
Note that an oblong is a rectangle that is not a square. It has two pairs of parallel sides of different lengths, and four right angles.
 Whakarara rite gets so excited by all the different diamondlike shapes that he folds in to a particular diamond shape in ahutoru (that is in 3 dimensions) called a koeko tapatoru (a triangular pyramid or tetrahedron). If you have a set of polyhedra, you will find a collection of pyramids. If not, locate a picture of one online).
 By folding the whakarara rite out and in, turn him into a koeko tapatoru.
 The following diagram illustrates the change to a triangular pyramid.
Koeko tapatoru is not a taparau (polygon) because he is no longer flat. He is three dimensional (ahutoru).

How would you describe his appearance now?
You might introduce some simple language of polyhedra, such as faces (flat surfaces), edges, and vertices (corners).
Triangular prism to hexagon
 Koeko tapatoru stays like this for only a short time before changing back into a whakarara rite (rhombus). A little while later, Whakarara rite stops in at his favourite food place called Geo Flatworld Takeaways to get a kai of tapaono rite (regular hexagonal shape) chips.
 Whakarara rite wants to fold into a tapaono rite (regular hexagon).
Find a way of folding to make a tapaono rite that is made out of six equilateral triangles (e ono, tapa toru rite).
 The following diagram illustrates the change to a tapaono rite.
 Ask the students to check that it is a regular hexagon:
Are all the sides equal? How do you know?
Are all the angles equal? How do you know?
What rotational and reflection symmetry does a tapaono rite have? (Six lines of reflective symmetry and rotational symmetry of order six)
Session 5
In the final session we reflect on the shapes that we have explored during the week.
 At home Tapaono rite feeling tired after a long day, sits on the floor, closes his weary eyes, curls up into the frustum of a triangular pyramid and dreams about all the different shapes he was able to make ... with your expert help of course!
 Tell the students that a frustum of any solid shape is made by making a cut parallel to the base and removing the top of the solid. Show them the example of the cone.
 Next get the students to explore how they might ‘curl’ their tapaono rite (regular hexagon) up so that they can make such a frustum.
 Pose the following problems:
Name the different shapes of the faces of your frustum.
Make some statements about some of the attributes of the shapes that go to make up the frustum.
If Tapaono rite had sides of length two, how long were the sides of the open top and the base of the frustum?
Teaching notes:
The following diagram illustrates how to change a hexagon into the frustum of a triangular pyramid.
Fold B to O, then A to O, then C to O to make a triangle shape. Lift the flaps so D and E touch, F and G touch and H and I touch.
The open top has sides of length two because these sides are the same as Tapaono rite’s sides. The base sides are of length three. You can find this by measuring. (If you are clever you can use right angled triangles to work it out without measuring.)
Home Link
Dear family and whānau,
This week our class has been working on polygons. We’ve talked about them in both Māori and English. Ask your child to describe a porowhita, a parallelogram, a tapawhā and a taparara to you.
Problems to work on
 Is it possible for a taparara to have two equal sides?
 Can you make the frustum of a square pyramid?
Note: A frustum of any solid shape is made by making a cut parallel to the base and removing the top of the solid (see cone example).
Te whānau taparau  the polygon family
This unit examines the properties of polygons and how these are related. It also gives the names in both Māori and English.
This unit allows students to develop an understanding of the geometrical features of polygons and how classes of polygons are defined. It also aims to develop aspects of symmetry (reflective and rotational) through a problem solving approach.
A polygon is a planar (flat) shape that is bounded by straight sides. The relationships between sides and angles are used to create classes (groups) of shapes. In this unit students form quadrilaterals and triangles. Each class of shapes contains subclasses. For example, a quadrilateral might have pairs of opposite parallel sides. If a quadrilateral has only one pair of parallel sides, it is called a trapezium. If it has two pairs parallel sides, it is called parallelogram. Some parallelograms have internal angles that are right angles. That class of shapes is called rectangles. Definition, and reasoning with those definitions, is an important feature of geometric thinking.
In this unit, mathematical language is also explored particularly in terms of te reo Māori. It is envisaged that such an exploration will give rise to descriptions that can incorporate both languages, to support students to make sense of defining properties of 2dimensional and 3dimensional shapes.
The unit begins with string geometry to set the scene for investigating shapes and their properties using folding and possible turning techniques. Activity progresses to an examination of regular polygons where Māori terms are introduced. The concept of whānau or family is introduced to reinforce the fact that polygons are linked in a range of ways.
For this unit you will need to know, and the students will need to find out, the following:
porowhita = circle
whānau = family
taparau = polygons
tapatoru = triangle
tapawhā = square
tapawhā whakarara = parallelogram
whitianga = diameter
pūtoro = radius
paenga = circumference
tapatoru rite = equilateral triangle
puku = tummy
e toru, nga tapatoru rite = made from 3 smaller equilateral triangles
tapawhā whakarara rite = a rhombus
taparara = trapezium
tapawhā rite = square
tapawhā hāngai = an oblong
koeko tapatoru = a triangular pyramid
ahu3 = 3 dimensions
tapaono rite = regular hexagon
kai = food
e ono, tapatoru rite = 6 equilateral triangles
The learning opportunities in this unit can be differentiated by providing or removing support to students, or by varying the task requirements. Ways to support students include:
Tasks can be varied in many ways including:
The contexts for this unit can be adapted to suit the interests and cultural backgrounds of your students. Capitalise on the interests of your students. Shape, especially those with symmetry, is a commonality across most human cultures. Look for shapes and symbols that embody the target two dimensional shapes. For example, tukutuku panels use the symmetries of shapes in design. Three dimensional shapes are common in the environment and in human construction. Whare are usually the shape of a prism and the properties of prisms give them structural strength. Pyramids are well known to students through studies of Egypt and Central American civilisations, and are common in playgrounds and modern architecture.
Getting Started
In this session we use loops of string or elastic to form shapes according to given requirements. We then convince others that our shape meets the requirements.
Use the members of your group and the loop of string to make a shape that has four equal sides.
How did you find each of the equal length sides?
Using the string, how might you show any lines of reflective (line) symmetry?
How might you show whether the corners are right angles (without using a protractor)?
Convince another team member that your shape is correct.
What is the mathematical name for your shape?
Most groups will create squares, but some groups may create rhombi (rhombuses).
What properties must a shape have to be a square? (Four equal sides, four right angles, two pairs of parallel sides)
Are there other shapes that have four equal sides?
What are those shapes called? (Rhombi)
What are the properties of a rhombus? (“Four equal sides” is the only defining property. Note: Opposite angles are equal so there are two pairs of equal angles.)
Is a square a kind of rhombus? (Yes. It has four equal sides).
Turn various shapes so they appear differently, and ask:
Is this still a square/rhombus? Why? (Orientation does not effect the properties of the shape.)
Use the members of your group and the loop of string to make a shape that has four sides, two of which are parallel.
You may need to define parallel first before groups start work. Look up a definition online. Railway tracks are a good metaphor.
Note: In the US, a trapezium refers to a quadrilateral with no parallel sides, and some definitions of trapezium refer to at least one pair of parallel sides, therefore including parallelograms.
Is a rectangle a special parallelogram? (Yes, all rectangles have two pairs of parallel sides, so they are all types of parallelograms).
Is a trapezium also a type of parallelogram? (A trapezium has only one pair of parallel sides, so it is not a parallelogram).
Can you form triangle with an angle greater than ninety degrees? Try. (Yes, e.g. 120⁰, 35⁰, 25⁰)
What is true about the interior angles of an equilateral triangle? (Equal angles)
Can a triangle have one right angle? (Yes) Make a triangle like that.
Can a triangle have two right angles? (No)
Session 2
Over the next few sessions, students create Te Whānau Taparau (The Polygon Family) by transforming a circle.
The Circle
What makes a circle different from polygonal shapes, for example the triangle (tapatoru), square (tapawhā) or the parallelogram (whakarara)?
Students should note that polygons have straight sides. In fact, a polygon is a 2dimensional shape that is bounded by three or more straight sides.
What are some names that are used to describe parts of a circle?
It is important for students to consider the idea of a circle as a locus, an infinite set of points that meet a condition or set of conditions. Take your students outside with skipping ropes and chalk. Challenge them in teams of three or four.
Use your rope and chalk to create a perfect circle.
The defining property of a circle is that it is a set of points the same distance from the centre. The length of the rope is the radius of the circle.
The relationship between pae and rangiwhitu is an extension or extra investigation, usually reserved for later levels. Pi (3.14…) is the ratio of circumference to diameter, that is the circumference equals the diameter multiplied by pi.
Find the centre of your paper circle. What methods can you find?
Folding the circle in half in two different orientations creates an intersection of diameters. The point of intersection is the centre.
The circle becomes an equilateral triangle
Porowhita looked in the mirror one day and decided he was getting bored of being a circle. Porowhita decided that he wanted to look just like tapatoru rite. He needs your help.
Using the circles (provided or made), fold Porowhita to make the biggest possible tapatoru rite (equilateral triangle). You may need to experiment with different ways of folding to get the largest one possible.
Develop a method that you will use in order to check that the shape you have made satisfies the requirements needed for a tapatoru rite, and that it is the largest equilateral triangle possible.
How many lines of reflective symmetry does the triangle have?
Where are the lines located?
Does the triangle have rotational (turn) symmetry?
Why must the internal angles be equal for the shape to have rotational symmetry?
What other types of triangle can be folded from a circle?
What happens if you do not fold into the centre of the circle?
Is a rightangled triangle possible? (One side to the triangle must be the diameter)
Session 3
Triangle to trapezium
Here is Hannah's work. She has sorted some quadrilateral shapes into two different categories: trapezia and not trapezia.
Trapezia
Not Trapezia
Hannah states that her sorting procedure is based on the relationship between one pair of sides. If just one pair of sides is parallel, then it’s a trapezium. The other pair of sides cannot be parallel.
Do the parallel sides have to be different lengths? Why? (If the side lengths are equal then the other pair of sides will be parallel. The quadrilateral will be a parallelogram not a trapezium)
Ask questions that encourage the students to focus on quadrilaterals that have one pair of parallel sides. Some students might observe that a trapezium often looks like a triangle with the top cut off leaving an edge that is parallel to the side opposite.
Explore how you might fold Tapatoru rite to make a taparara (trapezium).
The trapezium must contain three equilateral triangles within it, that are the same size.
Let students work on the challenge in pairs.
Using the taparara (trapezium) that you have made:The following diagram illustrates the change to a trapezium.
Trapezium to rhombus
Write two sentences that describe Whakarara rite’s appearance.
What fraction of the original tapatoru rite (equilateral triangle) is Whakarara rite? (One half)
Session 4
Rhombus to a triangular pyramid
Note that an oblong is a rectangle that is not a square. It has two pairs of parallel sides of different lengths, and four right angles.
Koeko tapatoru is not a taparau (polygon) because he is no longer flat. He is three dimensional (ahutoru).
How would you describe his appearance now?
You might introduce some simple language of polyhedra, such as faces (flat surfaces), edges, and vertices (corners).
Triangular prism to hexagon
Find a way of folding to make a tapaono rite that is made out of six equilateral triangles (e ono, tapa toru rite).
Are all the sides equal? How do you know?
Are all the angles equal? How do you know?
What rotational and reflection symmetry does a tapaono rite have? (Six lines of reflective symmetry and rotational symmetry of order six)
Session 5
In the final session we reflect on the shapes that we have explored during the week.
Name the different shapes of the faces of your frustum.
Make some statements about some of the attributes of the shapes that go to make up the frustum.
If Tapaono rite had sides of length two, how long were the sides of the open top and the base of the frustum?
Teaching notes:
The following diagram illustrates how to change a hexagon into the frustum of a triangular pyramid.
Fold B to O, then A to O, then C to O to make a triangle shape. Lift the flaps so D and E touch, F and G touch and H and I touch.
The open top has sides of length two because these sides are the same as Tapaono rite’s sides. The base sides are of length three. You can find this by measuring. (If you are clever you can use right angled triangles to work it out without measuring.)
Dear family and whānau,
This week our class has been working on polygons. We’ve talked about them in both Māori and English. Ask your child to describe a porowhita, a parallelogram, a tapawhā and a taparara to you.
Problems to work on
Note: A frustum of any solid shape is made by making a cut parallel to the base and removing the top of the solid (see cone example).
Figure It Out Links
Productive thinking
This unit explores situations that involve multiplication and division using equal sets and rates. Students learn to apply the properties of whole numbers under multiplication, to derive new answers for basic multiplication and division facts.
The simplest form of multiplication problem involves finding the total of a given number of equal sets. Consider this problem:
There are eight cartons of eggs. Each carton contains four eggs.
How many eggs are there altogether?
The problem can be represented mathematically as 8 x 4 = □. Eight represents the number of sets (the multiplier). Four is the number in each set (multiplicand) and represents the unit rate of “four eggs per carton.” The x symbol represents “of” in the sense of connecting eight sets of four. The empty box is the product or total and the equals sign represents sameness of quantity or balance.
Division with equal sets takes two forms depending on which factor is unknown. Sharing division comes for equally distributing a total number of objects, the dividend, into a given number shares (the divisor), which results in an amount per share (the quotient). For example:
There are 32 eggs and eight cartons of the same size.
How many eggs go into each container?
Note that 32 ÷ 8 = 4 represents the sharing of 32 (the dividend) into 8 equal sets (the divisor) which results in a quotient of “4 eggs per carton.” Division also applies to measurement contexts such as:
There are 32 eggs. Four eggs go into each carton.
How many cartons are needed?
Note that the rate is known, “4 eggs per carton”, and that becomes the unit of measure. “How many fours are in 32?” answers the problem. That can be written as 32 ÷ 4 = 8.
Both equal sharing and measuring problems are common in the real world. Developmentally, students tend to build up solutions to these problems using addition at first, progressing to multiplication. With appropriate opportunities to learn, students later come to treat division as a separate but related operation.
The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:
The context used for this unit include sports, packaging food, playing games and gardening to provide everyday situations that students are likely to be interested in. You may wish to change the contexts to situations more relevant to your students’ everyday lives, interests, or cultural identities. For example, cookies in cartons might become kumara or corn in bins at a hāngi, sports might become kapa haka events, and gardening might be related to the local community garden. Encourage students to be creative by accepting a variety of strategies from others and asking students to create their own problems for others to solve, in contexts that are meaningful.
Session One
Begin each session in this unit with an exercise in deriving from basic facts. Use a Slavonic Abacus to show how many different facts can be found from a given basic fact. Preferably use a physical abacus though virtual forms of the abacus are available online.
For example:
Why are the answers to 6 x 5 and 3 x 10 the same?
Students should notice that two fives make one ten, which explains the doubling and halving effect.
2 x 5 = 10
4 x 5 = 10
6 x 5 = 10, etc.
What does 0 x 5 equal? What does 10 x 5 equal? 12 x 5 equals?
You might encourage your students to remember the products if they do not already know them. Recite the facts in order, 1 x 5 =5, 2 x 5 = 10, 3 x 5 = 15, …etc. Erase two facts randomly each time and see if your students can recall the facts. Move to calling out particular facts for them to provide the product, e.g. “Seven multiplied by five.”
If we know six multiplied by five equals 30, what other facts could we work out?
Some examples might be:
5 x 5 = 25 7 x 5 = 35 6 x 4 = 24
6 x 6 = 36 6 x 7 = 42 12 x 5 = 60
For each derived fact, show clearly how 6 x 5 is altered by moving beads on the abacus (as shown above).
Tell them to create a spider’s web of facts that can be found using 8 x 5. Ask them to make as many connections as they can.
For example:
Share the students’ ideas and create a large whole class spider’s web of connections.
Mere has 60 kumara seedlings.
She wants to space the plants out in straight rows.
How many rows can she make?
How many seedlings will go in each row?
How many different answers can you find?
Can you find all the possible answers?
Solutions like 12 + 8 + 12 + 8 + 12 + 8 and 7 + 5 + 6 + 6 + 8 + 4 + 9 + 3 + 10 + 2 are acceptable (the addends are the number of seedlings in each row). These strategies usually have a system involved. For example, the last set of addends seems to be based on 12 (7 + 5, 6 + 6, etc.). Concentrate on solutions in which the numbers of plants in a row are equal. As students offer solutions organise your recording like this:
Students might try out three rows, four rows, seven rows etc. to see if that number of rows will work. A calculator might prove useful to find missing solutions. You could discuss simple divisibility rules, e.g. the digits of numbers divisible by three always add to a multiple of three, 54 ÷ 3 = 18 (divisible) and 5 + 4 = 9 (a multiple of three).
Division is efficient while multiplication requires a bit of trial and error, e.g. 60 ÷ 8 = 7.5 which is preferable to 8 x 6 = 48, 8 x 7 = 56, 8 x 8 = 64,… Aim to get a full set of factors for 60. Note that each pair of factors, such as 5 x 12, gives two solutions to Mere’s problem, e.g. 5 rows of 12 plants or 12 rows of 5 plants.
After Mere has planted her 60 seedlings Uncle Rewa gives her 12 more.
“That works out okay,” she says, “I’ll just put more plants in each row.”
Which of the previous solutions will still work if 12 plants are added?
Do they recognise for a solution to be built on, at least one existing factor must be a factor of 12?
For example, 2 rows of 30 plants cannot be changed to “3 rows of …” but it can be changed to 2 rows of 36 plants. 5 rows of 12 plants can be changed to 6 rows of 12 plants.
Come up with a list of solutions; 1 row of 72, 2 rows of 36, …, 6 rows of 12.
Are there any ways to plant 72 kumara seedlings in equal rows that are not on our list?
Students might identify 8 rows of 9 and 9 rows of 8 but these options cannot be created from the solutions for 60 plants without transplanting.
Session Two
6 x 10 = 60 so 6 x 9 = 54 (Six less)
Can you find patterns in the set of nine times facts?
1 x 9 = 9
2 x 9 = 18
3 x 9 = 29
…
9 x 9 = 81
10 x 9 = 90
Students might notice these patterns:
In the products the tens digits go up by one while the ones digits go down by one.
The multiplier is always one less that the tens digit in the product, e.g. In 8 x 9 = 72, 7 is one less than 8.
The digits in the product always add to nine, e.g. 4 x 9 = 36, 3 + 6 = 9.
Will eight plates work? Why? Why not?
24 ÷ 8 = 3, 36 ÷ 8 = 4 r4 or 4 ½, 48 ÷ 8 = 6. Therefore, eight plates will only work if some cookies are not used or are cut into halves. One strategy breaks a condition and the other will ruin the presentation of each plate.
Two plates will work because 24, 36 and 48 divide equally by two. Each plate might be too big to sell.
Three plates will work because 24, 36 and 48 divide equally by three. Each plate might be too big to sell.
Will five plates work?Why? Why not?
Five plates will not work as 24, 36, and 48 are not divisible by five (not in the five times table).
Will six plates work? What about seven plates? Etc.
Some students might list all the factors of 24, 36, and 48 to see what numbers of plates might work. You might give the students the start of the table and ask them to complete it.
Number of Plates
1
2
3
4
5
6
7
8
9
10
11
12
24 raisin
●
●
●
●
●
●
●
48 oatmeal
●
●
●
●
●
●
●
36 chocolate
●
●
●
●
●
●
●
Discuss: Should we stop at 12 plates? Are there other numbers of plates that might work?
If we use six plates, how many of each cookie is on each plate?
If we use twelve plates, how many of each cookie is on each plate?
Can we work this out from the six plate answer? How?
Six plates with 4 raisin, 8 oatmeal, and 6 chocolate cookies is probably the most practical.
A double number line of Nico’s hike looks like this:
Is there an easier way to predict the amount of time Nico will take?
Students might see that multiplying the time, in hours, by three gives the distance that Nico hikes, in kilometres. 8 x 3 =24 so it takes Nico over eight hours. Some students might be able to get more precise.
If Nico hikes 3 kilometres each hour,how long will it take him to hike 1 kilometre? (1/3 of an hour or 20 minutes)
Build up the time and distance pairs in a similar way, (1,7), (2,14), (3,21),... Can students see that multiplying the time, in hours, by seven gives the distance, in kilometres?
Can they be more precise than saying that “Kat takes over 8 hours to cycle 48 kilometres”? (two kilometres takes her 2/7 of an hour. That’s less than one third and equals about 17 minutes.)
Niko is hiking a 30 kilometre track.
He can walk at about 4 kilometres per hour, including rest stops.
Kat is cycling a 45 kilometre trail .
She rides at about 6 kilometres per hour, including rest stops.
Session Three
2 x 3 = 6 4 x 3 = 12 8 x 3 = 24
What patterns can you see?
Students should notice that the product doubles as the multiplier doubles.
What multiplication fact would come next?
(16 x 3 = 48. You might need two abacuses to show that)
2 x 6 =12 4 x 6 = 24 8 x 6 = 48
What patterns can you see?
2 x 4 = 8, so 4 x 4 = ____, so 8 x 4 = ____, so 16 x 4 = ____
2 x 8 = ____, so 4 x 8 = ____, so 8 x 8 = ____, so 16 x 8 =
7 x 2 = ____, so 7 x 4 = ____, so 7 x 8 = ____, so 7 x 16 =
2 x 11 = ____, so 4 x 11 = ____, so 8 x 11 = ____, so 16 x 11 =
Note that a simplified form of the game is available using this Figure It Out task: Dicey Dabble.
Session Four
Use this example first:
4 x 5 = 20 4 x 6 = 4 x 7 =
8 x 5 = 40 8 x 6 = 8 x 7 =
Can you make your spider web even larger?
Suppose we decide to run a sports day.
Each sport is on at a particular time of the day.
No other sport is played at that time. It is too hard to organise.
We want as few players as possible that are reserves at any time. Everyone must be in a team.
What is the ideal number of entries (people) for the sports day?
Explain your answer.
How many players will we need at the Sports Day if we include these sports as well?
This task is very challenging but students will have some idea of how to approach it from the earlier problem. Make calculators available as the focus is on thinking with multiples, not on calculation.
The easiest solution is to multiply 40 by nine. 360 is divisible by 8, 10 and 9, which is required for there to be no reserves at anytime.
Session Five
Dear parents and whānau,
In mathematics this week, our focus has been on multiplication and division. We learned to work out new multiplication facts from facts we already know. To show how that works, ask your student which facts could be worked out from 8 x 5 = 40.
He or she might say that 8 x 5 and 4 x 10 have the same answer. They might say that 8 x 7 must be 16 more than 40 and explain why. There are lots of other connected facts.
Now is a good time for your student to learn their basic multiplication facts at home and school. You might like to find out what facts they know and help them to learn the facts they do not.
Boxing On
This unit supports students to develop their ideas about capacity using standard units.
Volume is the measure of space taken up by a threedimensional object. The space within a container is known as its capacity but as the thickness of many containers is negligible, it has become acceptable to refer to the space inside a container as volume too.
In this unit students find the capacity of containers using cubic units (cubic centimetres, millilitres, litres and cubic metres), and explore relationships between these measures. By constructing containers of a given volume students strengthen their understanding of standard units.
The learning activities in this unit can be differentiated by varying the scaffolding provided to make the learning opportunities accessible to a range of learners. Students who need additional support with measuring capacity can use the materials described for each session to “fill up” the boxes, then count the number of measures used. Students who need less support will be able to calculate volumes from measured lengths.
This unit can be adapted to suit the experiences of your students. It uses boxes, and describes the use of small boxes from food and household items such as sugar cubes, toothpaste, cocoa, and spices. Use any kind of rectangular box or container that is available, and that students are familiar with. Examples include paper bags, takeaway containers, and small gift boxes. Prior to teaching the unit you may like to source a collection of boxes for students to share. One way to do this would be to ask students to bring boxes from home, or search for suitable boxes around the school.
The resources needed for each session are listed alongside each activity below.
Session 1: Sugar Boxes
In this session we design boxes to hold 64 sugar cubes.
Resources
Possible boxes include:
2cm x 2cm x 128cm for 64 cubes in a single row
4cm x 2cm x 64cm for 2 rows of 32 cubes
8cm x 4cm x 16cm for 2 layers, each with 4 rows of 8 cubes
Session 2: Toothpaste boxes
In this session we explore the size of commercial boxes and construct a rectangular box (cuboid) of a given size.
Resources
The students may choose to work this out by filling each box with placevalue cubes, but is there an easier way?
Session 3: Box capacity
In this session we find the capacity of boxes in millilitres and cubic centimetres.
Resources
Session 4: The metre cube
In this session we find the number of placevalue blocks that fill a metre cube.
Resources
Session 5: Air space
In this session we investigate the capacity of the classroom.
Resources
Family and whānau,
How many children from our class can fit in a cubic metre?
This week at school we have been exploring the capacity of objects and have used both litres and cubic metres and centimetres as our measuring units.
We invite you and your family to guess how many children from our class will be able to squeeze into a cubic metre. We will be trying this out on Friday so please send your guess to school before then.
Family/Whanau Name:___________________
We think that _____________________children will be able to squeeze into the cubic metre. This is a diagram of our thinking that went into our guess.
Figure It Out Links
Some links from the Figure It Out series which you may find useful are:
Food for thought: Using equations
The purpose of this unit is to support students to represent number problems as equations, and to strategically choose the best operation to solve problems in context.
Singlestep finding of unknowns can be classified in this helpful way:
Operation
Result unknown
Change unknown
Start unknown
Addition
4 + 6 = [ ]
4 + [ ] = 10
[ ] + 6 = 10
Subtraction
10  6 = [ ]
10  [ ] = 6
[ ]  6 = 4
Multiplication
3 × 5 = [ ]
[ ] × 5 = 15
3 × [ ] = 15
Division
15 ÷ 3 = [ ]
15 ÷ [ ] = 5
[ ] ÷ 3 = 5
Using this classification is it easy to develop story contexts for problem solving. For example, 4 + [ ] = 10 might be framed as “Sid has 4 apples. He picks some more apples and now he has 10 apples. How many apples has he picked?” The problem could be solved as subtraction, 10  [ ] = 4, but the context is one of joining sets of apples.
Varying the location of the unknown substantially changes the difficulty of the problem, assuming the numbers are similar. Change unknown problems, such Sid’s apple picking, involve considering the possibilities for change. Start unknown problems require inverse thinking since there is no beginning state. For example, “Sid has some apples. He picks six apples and now he has 10 apples. How many apples did Sid have to start with?” Changing the operation to subtraction, 10 – 6 = [ ], involves reconceptualising the role of the whole and parts. Change unknown problems tend to be the most difficult, assuming similar numbers and the same operation.
The learning opportunities in this unit can be differentiated by providing or removing support to students, by varying the task requirements. Students need to focus on the decisionmaking process of which operation/s to use. While the development of calculation strategies can be facilitated through solving the problems, that is not the primary purpose. Ways to support students include:
Tasks can be varied in many ways including:
The contexts for this unit can be adapted to suit the interests and cultural backgrounds of your students. A menu from a school cafeteria, or tuck shop, and a fruit and vegetable shop provide the contexts of the unit. You may need to show students a video clip of lunch schemes in larger schools or overseas to help them appreciate the usefulness of the context. If the foodbased contexts are culturally inappropriate to your students change the items to those that match the everyday situations students regularly encounter, e.g. dollar shop, school gala, toy shop, etc.
Session 1
In this session students use the clues provided to find out the prices of items on a menu. The clues involve applying addition and subtraction with unknowns.
Here is the menu from the Cafeteria at Kiwi School.
What is a cafeteria?
What is meant by daily specials on this menu? (A cycle of options available on days of the week)
Explain what is meant by Regular Daily Items? (Items that are always on sale, no matter the day of the week)
Assume that students know what is meant by snacks and drinks.
Give them some time to decide on their preferences and to record those preferences.
How much will you pay for your lunch today?
Students may notice that the prices are not visible.
Luckily we have a set of clues to help us work the prices. Before we start can someone clarify what a combo is.
Students will know that combo is short for combination.
What is a combination?
What are the possible options for one choice and three choices?
How could we organise those possibilities?
Ask students to write some possibilities on their copy of the menu, e.g. $1.00 and $2.00, $2.00 and $4.00, $3.00 and $6.00, etc.
Which possibilities are unlikely?
One choice must be cheaper than two choices otherwise people would not buy one choice.
Must the prices be whole dollars? What does that mean for the possibilities?
Since there are a huge range of possibilities it might be better to write a rule, like 2 x, to show the relationship. Watch to see how your students record their rule.
If two daily items cost $5.00 then each item must cost $2.50.
$2.50  $1.00 = $1.50 is the cost of one snack and $2.50 – $0.50 = $2.00 is the cost of one beverage.
3 x $2.00 = $6:00 so the cost of three choices equals $6.00
What is the cost of one choice?
Record [ ] x 2 = $6.00 so [ ] = ? ($3.00  cost of one choice)
Combo 1: $6.50 Combo 2: $7.50 Combo 3: $8.00 Combo 4: $3.00
Remind them to order the lunch they chose and calculate the cost of it.
Was it cheaper for you to order a Combo?
For example, Combo 2 and Combo 4 cost a total of $10.50.
How did you record that clue?
Students might have drawn arrows connecting the Combos on the menu, or have other strategies.
Record: C2 + C4 = 10.5
What might that equation mean?
I could record the same relationship as ○ + □ = 10.5
Which way do you think is better?
The key point is that the meaning of the symbols needs to be clear. It is good practice mathematically to define what both symbols represent.
How might you record the clues, “One snack costs $1.00 less than one daily item, and one beverage costs 50 cents less than one daily item”?
Students might invent notation like; s = d  1 and b = d – 0.5 or use ‘empty’ shapes instead of the letters.
Session Two
In this session students explore change and start unknown addition and subtraction problems. They record their solution strategies as equations with specific unknowns and recognise when either addition or subtraction can be used to solve the problem.
The new price of regular daily items is $3.40.
What was the old price?
What is the price increase?
Let students solve the problem in pairs before sharing their strategies as a whole class.
Use words, equations, and empty number lines to record their strategies.
By change unknown addition:
“$2.50 add what equals $3.40?”
2.50 + [ ] = 3.40
Students are likely to find the change in incremental jumps, like this, with $3.00 used as a benchmark.
If no students use subtraction to solve the problem, you might model a solution as:
“$3.40 subtract $2.50 equals what?”
3.40  2.50 = [ ]
Which strategy is the most efficient for the problem? (Addition is much easier)
The new price of Combo 2 is $6.80.
What was the old price?
What is the price decrease?
This problem is a change unknown subtraction problem that can be represented as:
“$7.50 subtract what equals $6.80”
7.50  [ ] = 6.60
Students are likely to use incremental jumps to find the missing number (subtrahend).
Could the problem be solved using addition?
The use of change unknown addition looks like this:
“$6.80 add what equals $7.50?”
6.80 + [ ] = 7.50
Highlight the use of equations and empty number lines as recording strategies.
Answers:
Adding the new prices and applying that discount gives:
3.70 + 3.40 + 0.80 + 1.40 – 1.50 = $7.80
Other answers are possible.
Session Three
In this session students represent change and start unknown problems with multiplication and division. They do so in the context of the Kiwi School Cafeteria used in the previous two lessons. The focus in this session is more on making sensible decisions about the operations to perform than on strategies for calculation. ‘Unfriendly’ amounts are used for the prices and students are required to choose appropriate operations to solve the problems.
The Manager sees that Room 7 ordered $17.00 worth of Regular Daily item 1, the stuffed potato. The class monitor has not written how many potatoes are ordered.
How can the manager work out how many potatoes to send to Room 7?
What information do we need to solve this problem? (The price of Regular Daily Items is $3.40 each)
What operation do we need to perform?
Students might realise that they need to answer “$3.40 multiplied by what number equals $17.00?”
How can this problem be written as an equation? ([ ] x 3.40 = 17.00)
Is there another operation that might be used?
What would the equation for that operation be? (17 ÷ [ ] = 3.40 ($17.00 shared into how many equal amounts equals $3.40?) or 17.00 ÷ 3.40 = [ ]($17.00 measured in amounts of $3.40 gives how many amounts?))
The second option for division gives a direct answer of five items. Let students use a calculator to find their answer then use five as the unknown in the other two equations.
5 x 3.40 = 17.00 (The calculator will not show whole number amounts with a decimal point)
17 ÷ 5 = 3.40
One day a single order came in from Room 2. The total amount was $25.90 and the form said “7x”. The manager knew that each student had ordered the same kind of item but what was it?
What does 7x mean on the order? (Seven, of the same item, are ordered)
What equations could we use to record the problem?
How could the manager work out what kind of item was ordered?
7 x [ ] = 25.90 25.90 ÷ 7 = [ ]
Use the equations to find out what kind of items were ordered.
Students might use the multiplication equation but that involves a trial and error process until the correct amount is found, e.g. 7 x 4.00 = 28.00, 7 x 3.00 = 21.00, etc. The division equation gives a direct answer. Substitute the answer back into the unknown in the multiplication equation to verify that both equations give the same result for the unknown, $3.70.
What type of item is this order for? (One Choice Daily Specials)
For example:
Room 12 sends in an order for 9 students who want to buy Combo 2 at $7.80 each.
The manager checks to see that the money inside the class envelope is correct.
How much money should there be?
Students will probably recognise that this is a result unknown multiplication problem.
What equations can you write to represent this problem?
9 x 7.80 = [ ]
How could we write the same problem as division?
[ ] ÷ 7.80 = 9 or [ ] ÷ 9 = 7.80
Discuss what each equation means, e.g. “The unknown total amount shared among the nine orders equals $7.80 for each order?”
Which way to record the problem is the most sensible?
Clearly the multiplication gives a direct answer, of $70.20.
Can students explain the use of numbers and operation in the problem they write?
In particular, observe that they understand the difference between sharing and measurement division.
Session Four
In this session students explore situations in which more that one operation is involved, and they learn to discriminate additive and multiplicative situations. To allow for more flexibility the context is changed from a canteen to a fruit and vegetable store.
Imagine you go to Pip’s Fruit and Vege Shop with $20.00 in total to spend. You must buy at least two different foods.
What will you buy?
How will you calculate the total cost?
What change will you get?
Record the operations you performed to get your total cost. We will try to figure out what you bought.
For example, a pair of students might record 1.5 x 2.79 + 3 x 1.99 + 2.4 x 3.70 = $19.04.
Students might work out that they bought 1.5kg of bananas, 3 avocados, and 2.4 kg of apples.
Tim and Iti spent a total of $17.50.
They bought 3.5 kg of mandarins and an amount of onions.
How many kilograms of onions did they buy?
An equation might look like this:
3.5 x 2.65 + [ ] x 2.75 = 17.50
How can we work out the amount that Tim and Iti spent on onions? (Subtract 17.50 – (3.5 x 2.65) = 8.23 (rounded))
What operation will tell us how many kilograms of onions they bought? (8.3 ÷ 2.75 = 2.99)
Tim and Iti bought about 3kg of onions.
An equation might look like this:
2.3 x 4.50  5 x 1.35 = [ ]
Notice that a difference is found by subtraction or adding on.
5 x 1.35 + [ ] = 2.3 x 4.50
In this case subtraction gives direct access to the difference of $3.60.
They bought 2 kilograms each of two different fruit. The total cost was $13.10.
What fruit did they buy?
An equation might look like this:
2 x [ ] + 2 x ( ) = 13.10
How can we find the total cost of 1 kilogram of each fruit? (divide by two)
[ ] + ( ) = 6.55
What two fruit prices match $6.55 per kilogram?
Students might realise that one price will need to have 5 in the hundredths place while the other must have 0 in the hundredths place. Only kiwifruit and plums will work.
How adventurous are they in combining the operations? (Do they stay in safe territory with addition and subtraction, or multiplication and division?)
Do they provide a solution to their problem showing the steps as equations?
In May apples cost $15 ÷ 3 = $5 per kilogram. Therefore, the price of kiwifruit is $5 + $1 = $6 per kilogram.
If Nashi pears cost more than plums, then mangoes cost 2.50 + 1.75 = $4.25 per kilogram than plums.
If plums cost more than Nashi pears, then mangoes cost 2.50  1.75 = $0.75 per kilogram than plums.
3.60 ÷ 2 = $1.80.
Dear parents and whānau,
In mathematics this week, our focus has been on writing and solving equations. Students are applying their understanding that an equation represents a balance, or sameness. The equals sign represents that two amounts on each side of it are the same, though the operations may look different and represent different operations.
Have your child explain their common sense understanding of these two problems, how they would solve the equation, and justify why their strategy works. Using a calculator is fine, provided your student can explain why they chose the operation they did.
Find the number that goes in the box to make each equation correct.
[ ]  27 = 51
7 × [ ] = 161
300 ÷ [ ] = 12
Listen to your student’s explanations, and ask questions if the strategies don’t make sense to you. Compliment them when their explanation is clear, and their solution is efficient and correct.
Thank you
Logo Licenses
This unit examines the use of reflective, rotational, and translational symmetry in the design of logos. Logos are designs associated with a particular trade name or company and usually involve symmetry to make them aesthetically pleasing as well as functional.
This unit centres on symmetry, particularly reflective and rotational symmetry, although there is some reference to translation symmetry. A shape has symmetry if it has spatial pattern, that is, it maps onto itself either by reflection about a line, or rotation about a point.
Consider the Mitsubishi logo. There are three lines where a mirror could be placed and the whole figure could be seen, with the image in the mirror forming the hidden half.
This logo also has rotational symmetry about a point. Each turn of 120⁰ (one third of one full rotation) maps the logo onto itself. Since the logo maps onto itself three times in a full turn of 360⁰, the figure has rotational symmetry of order three.
The mathematics of symmetry is found in decorative design, like kowhaiwhai in wharenui, and wallpaper patterns, and motifs such as logos. Human beings are naturally appreciative of symmetry, possibly it is prevalent in the natural world. Creatures are approximately symmetrical and reflections in water are a common example of mirror symmetry.
The learning opportunities in this unit can be differentiated by providing or removing support to students, by varying the task requirements. Ways to support students include:
Tasks can be varied in many ways including:
The contexts for this unit can be adapted to suit the interests and cultural backgrounds of your students. Capitalise on the interests of your students. Symmetry is common across all cultures of the world. Kowhaiwhai patterns on the rafters of wharenui (meeting houses), and designs on Fijian tapa or Samoan siapo cloth usually involve symmetries. Look for examples of symmetrical design in the local community. Encourage students to capture symmetric patterns they see and use the internet as a tool for finding images in Aotearoa. Search for symmetry to show how common geometric pattern is throughout the world.
Getting Started
What pattern can you see in the shape?
How did the way we made the shape affect the symmetry it has?
You may want to show how the whole shape is visible if the mirror is located along the fold line.
Mirror symmetry and fold symmetry have the same meaning with 2dimensional shapes.
It is important for students to see that one half of the shape maps onto the other half by folding and flipping actions. Each alternative term can be demonstrated. Holding a mirror along the fold line enables students to see an image of the whole shape even when one half is masked. Similarly, the amended (cut) circle can be refolded in half and traced around. Then flip the paper over the fold line and trace around it again. The traced figure will be that of the whole amended circle.
Before opening up the circle ask:
What pattern do you expect the shape to have?
Is there another type of pattern in this shape?
Students may not recognise that the cutout shape has rotational symmetry as well. Trace around the shape on a whiteboard. Turn the shape one half turn (180⁰) to show that the shape maps onto itself.
The shape has half turn symmetry. How many times will it map onto itself in a full turn?
Students should predict order 2 rotational symmetry. That means the shape maps onto itself twice, in a full turn.
How are the shapes the same? How are they different?
Do both shapes have the same symmetry?
Most students will realise that there are two lines of reflective symmetry (the fold lines) but the half turn rotational symmetry is harder to spot.
Session Two: Car logos
What make of car was that?
How do you know?
Session Three: Logos in the media
The Starbuck’s logo belongs in the bottom left cell as it has reflective symmetry but no rotational symmetry.
You work for an advertising company as a logo designer. There are four new companies that need new logos. They have stipulated that the logo must have some symmetry but must also suggest what goods and services they provide. (If you wish, they may also be required to come up with a slogan that captures the message, e.g. "Just do it".)
Here are the companies:
Session Four
Dear family and whānau,
This week we are looking at symmetry and at company logos. We appreciate your help by working with your child to find examples of logos in magazines, television commercials or on websites. For homework your child is asked to sketch two logos in their book and describe their symmetries. Ask your student to explain what reflective and rotational symmetries are, using the logos as examples.
Figure it Out Links
Some links from the Figure It Out series which you may find useful are: